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Contact three-manifolds with exactly two simple Reeb orbits

Daniel Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings and Hui Liu

Geometry & Topology 27 (2023) 3801–3831

It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus-one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example, the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation, the contact structure is universally tight, and in the case of the three-sphere the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid.

Reeb orbit, embedded contact homology, lens space, pseudorotation
Mathematical Subject Classification
Primary: 37J99, 53E50
Secondary: 53D42
Received: 29 September 2021
Revised: 24 January 2022
Accepted: 26 February 2022
Published: 5 December 2023
Proposed: András I Stipsicz
Seconded: Ciprian Manolescu, Leonid Polterovich
Daniel Cristofaro-Gardiner
Department of Mathematics
University of California, Santa Cruz
Santa Cruz, CA
United States
School of Mathematics
Institute for Advanced Study
Princeton, NJ
United States
Department of Mathematics
University of Maryland
College Park, MD
United States
Umberto Hryniewicz
RWTH Aachen
Michael Hutchings
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States
Hui Liu
School of Mathematics and Statistics
Wuhan University

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